Covariant quantum measurements which maximize the likelihood

Characterising a universal cloning machine by maximum-likelihood estimation

We apply a general method for the estimation of completely positive maps to the 1-to-2 universal covariant cloning machine. The method is based on the maximum-likelihood principle, and makes use of random input states, along with random projective measurements on the output clones. The downhill simplex algorithm is applied for the maximisation of the likelihood functional.

Optimal Trade-Off: Information Gain vs Distortion Loss In Finite-Dimensional Quantum Systems

In this thesis, I reflect on quantum instruments that measure the state of pure finite dimensional quantum systems. As the Heisenberg principle dictates, there exists a joint restriction to the information gain and distortion by measurement of a quantum system. I minimize the distortion of measurement and maximize the quality of the outcome by finding the optimal (covariant) instrument. In order to optimize this joint restriction, the family of covariant instruments is classified. This is an important side result.

Gradient-based stopping rules for maximum-likelihood quantum-state tomography

When performing maximum-likelihood quantum-state tomography, one must find the quantum state that maximizes the likelihood of the state given observed measurements on identically prepared systems. The optimization is usually performed with iterative algorithms. This paper provides a gradient-based upper bound on the ratio of the true maximum likelihood and the likelihood of the state of the current iteration, regardless of the particular algorithm used. This bound is useful for formulating stopping rules for halting iterations of maximization algorithms. We discuss such stopping rules in the c…

Verification of state and entanglement with incomplete tomography

There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numeric…

协变比特线

Covariant bit threads

本文将 HRT 协变全息纠缠熵公式重新表述为极小极大及 V-flow、U-flow 两类凸规划，并以线分布统一描述相关流。

连接量子层析成像与量子回溯

Connecting Quantum Tomography and Quantum Retrodiction

本文证明量子层析成像与量子回溯本质相同，Petz恢复映射等价于最大似然层析中的对数似然梯度更新，并推广到任意量子信道，建立了回溯、恢复映射与统计推断之间的直接桥梁。